TITLE: Existence of Open Covers
QUESTION [6 upvotes]: Do sets always have open covers exist? I know they are not always finite, but do infinite ones always exist?
I was reading baby rudin and the proofs for non-relative nature for compactness seems to require that. But I couldn't find any explanations on why I can assume that open covers always exist.

REPLY [4 votes]: One of the properties of open sets is that each point lies within at least one open set. If $A\subseteq X$, then for each $a\in A$ there is some open set $U$ such that $a\in U$, and in other words $A\cap U\neq\varnothing$.
So taking $\{U\subseteq X\mid U\text{ is open and } U\cap A\neq\varnothing\}$ is an open cover of $A$ by the fact given above.